Stability Analysis

While trying to solve an equation of the form A * y'' - B * Y + c = 0 using finite elements in a 3d hemishpere problem I find the surface contours plot becomes unstable for higher values of Coeff B. Is there anyway I can control this parameter and get decent results?.

The equation can be written as Y'' =(B/A) * Y - (C/A). If you use finite-difference for Y'' , you get (Y(i+1) -2Y(i) +Y(i-1)) / ((x(i+1)-x(i-1))/2)**2. Also Y can be written as Y(i). If we let dx2=((x(i+1)-x(i-1))/2)**2, then the final equation for Y(i) is Y(i)=(Y(i+1)+Y(i-1))/(2 + (B/A)*dx2) + (dx2*(C/A))/(2 + (B/A)*dx2 ). When the mesh size is small, dx2 is small. Y(i) is approximately equal to (Y(i+1)+Y(i-1))/2. So, as long as Y(i+1) and Y(i-1) are finite, the solution Y(i) ( the value between x(i+1) and x(i-1)) will be finite and bounded by Y(i+1) and Y(i-1). In this case is the average value of Y(i+1) and Y(i-1). For finite value of mesh size, dx2 will be finite ( not a very small value ). And for large value of B, the denominator (2+(B/A)*dx2) can become a large number. For example, for finite Y(i+1) and Y(i-1), the solution Y(i) can become very small or approach zero. In this case, Y(i-1) > Y(i) < Y(i+1). So a V-shaped profile is created. This result is artificial because the real solution should always close to the average value of the Y(i+1) and Y(i-1) by making mesh size very small, and thus very small dx2. So you should try to keep (B/A)*dx2 << 2, by making mesh size small. Or dx2 << (2/ (B/A)). I think, the finite element works in a similar way( with additional assumptions).

As this problem is in 3-d with varying material shape and thickness I could only reach an optimum mesh size. I would like to know if I can use some alogrithm to get over the wiggles created in my solution as I unable to further refine my grid ? natteri sudharsan